If either \(\pi _{Y\vert X}\)  is non-Gaussian or
\(\smash{\widehat{S}}\)  is nonlinear—e.g., if we seek a
non-Gaussian approximation of the prior
distribution—then
the linear ansatz for \(\smash{\widehat{\mathcal {T}}}\)  is no longer optimal.
In such cases,
we might want
to capture non-Gaussian
structure
by considering
parameterizations of
\(\smash{\widehat{\mathcal {T}}}\)  that depart
gradually and incrementally from that of
a linear function,
seeking
a balance
between accuracy
and computational cost
(see Section ).
In particular, we can regularize the
estimation of the rearrangement in
high dimensions using localization ideas similarMost of the discussion of Section 
is specific to a KR rearrangement whose
target distribution is a product measure on
\(\mathbb {R}^n\) —e.g.,
\(\mathcal {N}({\bf 0}, {\bf I}_n)\) —and thus does not apply to the estimation of \(\mathcal {T}\) .
See [1]} for additional
details.
to those of Section .
We propose to use a “banded” estimator
analogous to
(REF ) to leverage
decay of dependence in \(\pi \) , i.e.,
\(\widehat{\mathcal {T}}( z) = \left[\begin{array}{l}\widehat{\mathcal {T}}^1(z_{\mathcal {A}_{1,r}})\\[2pt]\vdots \\\widehat{\mathcal {T}}^{n}(z_{\mathcal {A}_{n,r}}) \\\end{array}\right], \) 
